---
date: 2022-04-03 20:34
modification date: Sunday 3rd April 2022 20:34:15
title: "homotopy groups of spheres"
aliases: [homotopy groups of spheres, Whitehead product, Whitehead square]

---

Last modified date: <%+ tp.file.last_modified_date() %>


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- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [stable homotopy group of spheres 1](stable%20homotopy%20group%20of%20spheres%201.md)

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# homotopy groups of spheres

![](attachments/Pasted%20image%2020220422211326.png)

# Finiteness theorem

See <https://people.math.wisc.edu/~maxim/spseq.pdf#page=29>
![](attachments/Pasted%20image%2020220403203429.png)

# Infinite order elements using the Whitehead product

![](attachments/Pasted%20image%2020220403203541.png)


# Unstable group computations

See <https://people.math.wisc.edu/~maxim/spseq.pdf#page=27>. Computes:

- $\pi_4 S^3$
- $\pi_4 S^2$
- $\pi_5 S^3$
- $\pi_{n+1}S^n$